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2018 | OriginalPaper | Chapter

8. Fourier Analysis of Periodic Weakly Stationary Processes

Author : Toru Maruyama

Published in: Fourier Analysis of Economic Phenomena

Publisher: Springer Singapore

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Abstract

During the decade around 1930, the world economy was thrown into a serious depression that nobody had previously experienced.

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previous chapter Spectral Representation of Unitary Operators

next chapter Almost Periodic Functions and Weakly Stationary Stochastic Processes

R. Frisch [6] also deserves special attention.

See Maruyama [22] for the outline of the theories of business fluctuations. See also Samuelson [24, 25], Hicks [9], Kaldor [11], R. Frisch [6], Keynes [18], Slutsky [28‐30].

Kawata [12, 13], Maruyama [20] and Wold [31] are classical works on Fourier analysis of stationary stochastic processes, which provided me with all the basic mathematical background. Among more recent literature, I wish to mention Brémaud [1]. Granger and Newbold [7] Chap. 2, Hamilton [8] Chap. 3 and Sargent [26] Chap. XI are textbooks written from the standpoint of economics.

See Kawata [17] pp. 5–7, for the proof. Loève [19] pp. 92–94 and Dudley [5] pp. 346–349 are also suggestive.

That is, both the real and imaginary parts are rationals.

We should use distinct notations for one-variable function ρ(u) and two variable function ρ(s, t). However, we use the same notation because there seems no possibility of confusion. ρ(u) is also called the autocovariance and ρ(u)∕ρ(0) is called the autocorrelation of the process.

In this evaluation, we are examining the situation where the upper limits (q, q′) of the sums tend to ∞, and the lower limits (p, p′) tend to −∞. So, without loss of generality, we may assume ${p', p<0,\; 0\leqq q, q'}$. There are various cases other than p′< p and q < q′. However, we can treat them in the same manner.

It is known that the right-hand side converges strongly. Hence we can specify the upper and lower limits of $\sum $ as p and −p. l.i.m. denotes the limit in $\mathfrak {L}^2$.

Crum [3].

It is well-known that

$$\displaystyle \begin{aligned} |\alpha -\beta|{}^2 \leqq 2(|\alpha |{}^2+|\beta|{}^2) \end{aligned}$$

for any $\alpha , \beta \in \mathbb {C}$, in general. Specifying α and β as

$$\displaystyle \begin{aligned} \alpha =e^{-t^2}X(t,\omega),\quad \beta=e^{-(t+u)^2}X(t+u,\omega), \end{aligned}$$

we obtain the desired result.

The graph of the function t↦e ^−2tu is depicted as follows according to the sign of u.

https://static-content.springer.com/image/chp%3A10.1007%2F978-981-13-2730-8_8/MediaObjects/417569_1_En_8_Figa_HTML.gif

(a)

In the case of u > 0, e ^−2tu monotonically converges to 1 at each t as u → 0. To say more in detail, it is monotonically increasing in the region t > 0, and decreasing in the region t < 0. Since we may assume $0<u\leqq 1$, without loss of generality, $e^{-2t^2-2tu}\leqq e^{-2t^2-2t}$ in the region t < 0. Hence applying the dominated convergence theorem and the monotone convergence theorem to the first and the second terms, respectively, of the right-hand side of the integral

$$\displaystyle \begin{aligned} \int_{\mathbb{R}} e^{-2t^2-2tu}dt = \int_{-\infty}^0 e^{-2t^2} e^{-2tu}dt + \int_0^\infty e^{-2t^2} e^{-2tu}dt, \end{aligned}$$

we obtain

$$\displaystyle \begin{aligned} \int_{\mathbb{R}} e^{-2t^2-2tu}dt\rightarrow \int_{\mathbb{R}} e^{-2t^2}dt \end{aligned}$$

(†)

as u → 0 (u > 0).

(b)

A similar argument applies to the case of u < 0, u → 0.

In general, assume that a numerical sequence {u _n} (u _n may either be positive or negative) converges to 0. Suppose

$$\displaystyle \begin{aligned} \int_{\mathbb{R}}e^{-2t^2}e^{-2tu_n}dt \nrightarrow \int_{\mathbb{R}}e^{-2t^2}dt. \end{aligned} $$

Then for sufficiently small ε > 0, there exists a subsequence $\{u_{n'}\}$ of {u _n} such that

$$\displaystyle \begin{aligned} \bigg|\int_{\mathbb{R}}e^{-2t^2-2tu_{n'}}dt-\int_{\mathbb{R}}e^{-2t^2}dt \bigg|\geqq \varepsilon \quad \text{for all}\; n'. \end{aligned} $$

If we choose a further subsequence $\{u_{n''}\}$ of $\{u_{n'}\}$ of the type (a) or (b), then we must have

$$\displaystyle \begin{aligned} \bigg|\int_{\mathbb{R}}e^{-2t^2-2tu_{n''}}dt-\int_{\mathbb{R}}e^{-2t^2}dt\bigg|\geqq \varepsilon \quad \text{for all}\; n''. \end{aligned} $$

A contradiction occurs.

Kawata [16], pp. 150–151.

X(t, ω) is strongly continuous by Theorem 8.5.

Let Φ ₁, Φ ₂, ⋯ be a sequence of Borel probability measures. Then there exist a certain probability space $(\varOmega , \mathbb {E}, P)$ and mutually independent random variables, the distributions of which are Φ ₁, Φ ₂, ⋯(Itô [10], p. 68).

Let X ₁, X ₂, ⋯ be independent real random variables and $g_1, g_2, \cdots : \mathbb {R}\rightarrow \mathbb {C}$ Borel measurable. Then Y ₁ = g ₁(X ₁), Y ₂ = g ₂(X ₂), ⋯ are independent random variables (Itô [10], p. 66). Based upon this result, Y (ω) and e ^−iZ(ω)t in the text are independent.

If we denote by $\hat {\nu }$ the Fourier transform of ν, we have

$$\displaystyle \begin{aligned} |\hat{\nu}(t+h)-\hat{\nu}(t)| &=\frac{1}{\sqrt{2\pi}}\bigg|\int_{\mathbb{R}}(e^{-i(t+h)x}-e^{-itx})d\nu(x)\bigg| \\ &\leqq \frac{1}{\sqrt{2\pi}}\int_{\mathbb{R}}|e^{-ihx}-1|d\nu(x) \\ &\rightarrow 0\quad \text{as}\quad h\rightarrow 0, \end{aligned} $$

by the dominated convergence theorem.

Hence if the right-hand side is a finite sum,

$$\displaystyle \begin{aligned}\xi\bigg(\bigcup_{j=1}^p S_j, \omega\bigg)=\sum_{j=1}^p \xi (S_j, \omega)\;\; \text{a.e.}\end{aligned}$$

According to Itô [10] p. 255, Kolmogorov’s important article was published in C.R. Acad. Sci. URSS, 26 (1940), 115–118. However, I have never read it, very regrettably. That is why I dropped it from the reference list.

See Maruyama [21] pp. 74–76.

$\nu _\xi (S)=\|\xi (S,\omega )\|{ }_2^2$ is a special case of Theorem 7.8(i) (p. 177).

The last equality is verified as follows. We consider a simple function $\varphi (\lambda )=\displaystyle \sum _{i=1}^n \alpha _i \chi _{S_i}(\lambda )\quad (S_i\cap S_j=\emptyset \quad \text{if}\quad i\neq j)$.

$$\displaystyle \begin{aligned} \int_{\mathbb{R}}\varphi(\lambda)dE(\lambda)X(0,\omega)=\sum_{i=1}^n\alpha_i E(S_i)X(0,\omega) =\sum_{i=1}^n \alpha_i \xi(S_i,\omega)=\int_{\mathbb{R}} \varphi(\lambda)\xi(d\lambda,\omega). \end{aligned}$$

The integration of e ^−iλt is a limit of such integrations of simple functions.

This problem was studied by Doob [4] Chap. X, §8, Chap. XI, §8 and Kawata [17] pp. 69–73. I try to clarify the subtle details embedded in their works. cf. Maruyama [23].

See Doob [4] Chap. II, §2.

The convergence of the series in (8.45) is in $\mathfrak {L}^2(\mathbb {T},\mathbb {C})$. However, the series is, actually, convergent a.e. and is equal to α(λ) according to Carleson’s theorem. cf. Carleson [2].

In the case $\mathbb {T}_0=\emptyset $ (and so α(λ) never vanishes), the discussion becomes much easier, since it is enough to define γ(S, ω) simply by

$$\displaystyle \begin{aligned} \gamma(S,\omega)=\int_S\frac{1}{\alpha(\lambda)}\xi(d\lambda,\omega) \end{aligned}$$

for any $S\in \mathbb {B}(\mathbb {T})$. Clearly, ν _γ(S) = m(S).

$$\displaystyle \begin{aligned} \mathbb{E}_{(\omega,\omega')}&|\gamma'(S,\omega,\omega')|{}^2 \\ =&\; \mathbb{E}_\omega\Big|\int_S\alpha_1(\lambda)\xi(d\lambda,\omega)\Big|{}^2 \\ &+\mathbb{E}_{\omega'}\Big|\int_S\alpha_2(\lambda)\eta(d\lambda,\omega')\Big|{}^2 \\ &+2{\mathbb Re}\mathbb{E}_{(\omega,\omega')}\int_S\alpha_1(\lambda)\xi(d\lambda,\omega)1(\omega')\int_S \overline{\alpha_2(\lambda)\eta(d\lambda,\omega')1(\omega)} \\ =&\int_S|\alpha_1(\lambda)|{}^2\nu_\xi(d\lambda)+\int_S|\alpha_2(\lambda)|{}^2\nu_\eta(d\lambda) \\ &+2{\mathbb Re}\mathbb{E}_{(\omega,\omega')}\int_{S\cap\mathbb{T}_+}\alpha_1(\lambda)\xi(d\lambda,\omega) \int_{S\cap\mathbb{T}_0}\overline{\alpha_2(\lambda)\eta(d\lambda,\omega')} \\ =&\; m(S\cap\mathbb{T}_+)+m(S\cap\mathbb{T}_0)+0. \end{aligned} $$

$$\displaystyle \begin{aligned} \int_{\mathbb{T}}&e^{-in\lambda}\alpha(\lambda)\gamma(d\lambda,\omega) \\ &=\int_{\mathbb{T}_+}e^{-in\lambda}\alpha(\lambda)\cdot\frac{1}{\alpha(\lambda)}\xi(d\lambda,\omega) +\mathbb{E}_{\omega'}\int_{\mathbb{T}_0}e^{-in\lambda}\alpha(\lambda)\cdot\alpha_2(\lambda)\eta(d\lambda,\omega') \\ &=\int_{\mathbb{T}_+}e^{-in\lambda}\xi(d\lambda,\omega)=\int_{\mathbb{T}}e^{-in\lambda}\xi(d\lambda,\omega). \end{aligned} $$

The final equality is justified by

$$\displaystyle \begin{aligned} \mathbb{E}_\omega&\Big|\int_{\mathbb{T}_0}e^{-in\lambda}\xi(d\lambda,\omega)\Big|{}^2=\int_{\mathbb{T}_0}\nu_\xi(d\lambda)\quad \text{(by D-K formula)} \\ &=\int_{\mathbb{T}_0}p(\lambda)dm=0\quad (p(\lambda)=0\; \text{on}\; \mathbb{T}_0). \end{aligned} $$

Since Z _n(ω) is a white noise, the covariance is given by

$$\displaystyle \begin{aligned}\mathbb{E}Z_{n+u}(\omega)\overline{Z_n(\omega)}= \begin{cases} 1\quad \text{if} \quad u=0,\\ 0\quad \text{if} \quad u\neq0. \end{cases} \end{aligned}$$

The convergence of $(1/\sqrt {2\pi })\displaystyle \sum _{k=-p}^qc_ke^{-ik\lambda }$ to C(λ) also holds good “almost everywhere” thanks to the Carleson theorem. Hence c _k is the Fourier coefficient of C(λ) corresponding to $(1/\sqrt {2\pi })e^{-ik\lambda }$.

The orthogonality, for instance, can be verified as follows. If S and $S'\in \mathbb {B}(\mathbb {T})$ are disjoint,

$$\displaystyle \begin{aligned} \mathbb{E}\theta(S,\omega)\overline{\theta(S',\omega)} &=\mathbb{E}\int_{\mathbb{T}}C(\lambda)\chi_S(\lambda)\xi(d\lambda,\omega) \int_{\mathbb{T}}\overline{C(\lambda)\chi_{S'}(\lambda)\xi(d\lambda,\omega)} \\ &=\int_{\mathbb{T}}|C(\lambda)|{}^2\chi_S(\lambda)\chi_{S'}(\lambda)d\nu_\xi=0\quad (\text{by D-K formula}). \end{aligned} $$

We can also establish

$$\displaystyle \begin{aligned} \frac{1}{\sqrt{2\pi}}\int_{\mathbb{R}}f(\lambda)g(\lambda)e^{-iz\lambda}dm(\lambda) =\frac{1}{\sqrt{2\pi}}(\mathbb{F}_2f\ast \mathbb{F}_2g)(z).\end{aligned}$$

(cf. Kawata [15] pp. 282–283.)

The third line of (8.56)

$$\displaystyle \begin{aligned} &=\frac{1}{\sqrt{2\pi}}\Big\{ \int_{\mathbb{R}_+} \alpha(\lambda-(t+u))\alpha(\lambda-t)dm(\lambda) +\int_{\mathbb{R}_0}\underbrace{\alpha(\lambda-(t+u))\alpha(\lambda-t)}_{(\dagger)}d\nu_\gamma\Big\} \\ &=\frac{1}{\sqrt{2\pi}}\Big\{\int_{\mathbb{R}}(\dagger)dm(\lambda)-\int_{\mathbb{R}_0}(\dagger)dm(\lambda) +\int_{\mathbb{R}_0}(\dagger)d\nu_\gamma\Big\}\\ &=\frac{1}{\sqrt{2\pi}}\Big\{ \int_{\mathbb{R}}(\dagger)dm(\lambda) -\int_{\begin{subarray}{l}\lambda \in \mathbb{R}_0 \\ \lambda-t\in \mathbb{R}_+ \end{subarray}} (\dagger)dm(\lambda) -\int_{\begin{subarray}{l}\lambda \in \mathbb{R}_0 \\ \lambda-t\in \mathbb{R}_0 \end{subarray}} (\dagger)dm(\lambda)\\ &\hspace{25mm}+\int_{\begin{subarray}{l}\lambda\in \mathbb{R}_0 \\ \lambda-t\in \mathbb{R}_+ \end{subarray}} (\dagger)d\nu_\gamma(\lambda) +\int_{\begin{subarray}{l}\lambda\in \mathbb{R}_0 \\ \lambda-t\in \mathbb{R}_0 \end{subarray}} (\dagger)d\nu_\gamma(\lambda)\Big\}\\ &=\frac{1}{\sqrt{2\pi}}\Big\{\int_{\mathbb{R}}(\dagger)dm(\lambda) -\int_{(\mathbb{R}_0-t)\cap\mathbb{R}_+} \alpha(\lambda'-u)\alpha(\lambda')dm(\lambda') \\ &\hspace{25mm} +\int_{(\mathbb{R}_0-t)\cap\mathbb{R}_+} \alpha(\lambda'-u) \alpha(\lambda')d\nu_\gamma(\lambda')\Big\}. \end{aligned} $$

The last two terms cancel out. So we obtain (8.56).

This section was added according to Professor S. Kusuoka’s suggestion. It is not included in the Japanese edition of this monograph.

For backward operators, see Granger and Newbold [7] Chap. 1.

Loève [19] p. 11.

Sargent [26] Chap. 11 and Shinkai [27] Chaps.7–8 are very suggestive.

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Title: Fourier Analysis of Periodic Weakly Stationary Processes
Author: Toru Maruyama
Publisher: Springer Singapore
Book: Fourier Analysis of Economic Phenomena
Print ISBN: 978-981-13-2729-2

Electronic ISBN: 978-981-13-2730-8

Copyright Year: 2018
DOI: https://doi.org/10.1007/978-981-13-2730-8_8

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